Given `f(t)=kt +1 and g(t)=3t+2`. If `fog=gof`, find k.

Options

A. 2
B. 3
C. 5
D. 4

Answer

To solve the problem, we need to find the value of \( k \) such that \( f(g(t)) = g(f(t)) \). We are given the functions:

- \( f(t) = kt + 1 \)
- \( g(t) = 3t + 2 \)

### Step 1: Calculate \( f(g(t)) \)

We start by substituting \( g(t) \) into \( f(t) \).

\[ g(t) = 3t + 2 \]

Now, substitute \( g(t) \) into \( f(t) \):

\[ f(g(t)) = f(3t + 2) = k(3t + 2) + 1 \]

Distributing \( k \):

\[ f(g(t)) = 3kt + 2k + 1 \]

### Step 2: Calculate \( g(f(t)) \)

Next, we substitute \( f(t) \) into \( g(t) \).

\[ f(t) = kt + 1 \]

Now, substitute \( f(t) \) into \( g(t) \):

\[ g(f(t)) = g(kt + 1) = 3(kt + 1) + 2 \]

Distributing \( 3 \):

\[ g(f(t)) = 3kt + 3 + 2 = 3kt + 5 \]

### Step 3: Set \( f(g(t)) \) equal to \( g(f(t)) \)

Now we have:

\[ f(g(t)) = 3kt + 2k + 1 \]
\[ g(f(t)) = 3kt + 5 \]

Setting them equal to each other:

\[ 3kt + 2k + 1 = 3kt + 5 \]

### Step 4: Simplify the equation

We can cancel \( 3kt \) from both sides:

\[ 2k + 1 = 5 \]

### Step 5: Solve for \( k \)

Now, we isolate \( k \):

\[ 2k = 5 - 1 \]
\[ 2k = 4 \]
\[ k = \frac{4}{2} = 2 \]

Thus, the value of \( k \) is \( 2 \).

### Final Answer

The value of \( k \) is \( 2 \).

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